Mathematical Statistics

Transcription

1 Mathematal Statsts 2 1 Chapter II Probablty 21 Bas Coepts The dsple of statsts deals wth the olleto ad aalyss of data Whe measuremets are tae, eve seemgly uder the same odtos, the results usually vary Varablty s a fat of lfe, ad proper statstal methods a help us uderstad data olleted uder heret varablty The term expermet refers to the proess of obtag a observed result of some pheomeo, ad a performae of a expermet s alled a tral of the expermet observato result, o a tral of the expermet, s alled a outome expermet, the outome of whh aot be predted wth ertaty, but the expermet s of suh a ature that the olleto of every possble outome a desrbed pror to ts performae, s alled a radom expermet The olleto of all possble outomes s alled the outome spae or the sample spae, deoted by S 22 lgebra of Sets If eah elemet of a set 1 s also a elemet of set 2, the set 1 s alled a subset of the set 2, dated by wrtg 1 2 If 1 2 ad 2 1, the two sets have the same elemets, dated by wrtg 1 2 If a set has o elemets, s alled the ull set, dated by wrtg Note that a ull set s a subset of all sets The set of all elemets that belog to at least oe of the sets 1 ad 2 s alled the uo of 1 ad 2, dated by wrtg 1 2 The uo of several sets, 1, 2, 3, s the set of all elemets that belog to at least oe of the several sets, deoted by or by f a fte umber of sets s volved The set of all elemets that belog to eah of the sets 1 ad 2 s alled the terseto of 1 ad 2, dated by wrtg 1 2 The terseto of several sets, 1, 2, 3, s the set of all elemets that belog to eah of the several sets, deoted by or by f a fte umber of sets s volved The set that ossts of all elemets of S that are ot elemets of s alled the omplemet of, deoted by I partular, S Gve S,, S, S, S S, ad ( ) Commutatve Law: B B ; B B ssoate Law: ( C ( B ; ( C ( B Dstrbutve Law: ( C ( ( B ; ( C ( ( B

2 Mathematal Statsts 2 2 De Morga s Laws: ; evet s a subset of the sample spae S If s a evet, we say that has ourred f t otas the outome that ourred Two evets 1 ad 2 are alled mutually exlusve f 1 2 Example 22-1: If the expermet ossts of flppg two os, the the sample spae ossts of the followg four pots: S {( H, H ), ( H, T ), ( T, H ), ( T, T )} If s the evet that a head appears o the frst o, the {( H, H ), ( H, T )} ad {( T, H ), ( T, T )} Example 22-2: If the expermet ossts of tossg oe de, the the sample spae ossts of the 6 pots: S {1, 2, 3, 4, 5, 6} If s the evet that the de appears a eve umber, the {2, 4, 6} ad {1, 3, 5} 23 Probablty Oe possble way of defg the probablty of a evet s terms of ts relatve frequey Suppose that a expermet, whose sample spae s S, s repeatedly performed uder exatly the same odtos For eah evet of the sample spae S, we defe #( to be the umber of tmes the frst repettos of the expermet that the evet ours The P (, the probablty of the evet, s defed by # ( lm How do we ow that # ( wll overge to some ostat lmtg value that wll be the same for eah possble sequee of repettos of the expermet? Oe way s that the overgee of # ( to a ostat lmtg value s a assumpto, or a axom, of the system However, t seems to be a very omplex assumpto ad does ot at all seem to be a pror evdet that t eed be the ase I fat t s more reasoable to assume a set of smpler ad more self-evdee axoms about probablty ad the attempt to prove that suh a ostat lmtg frequey does some sese exst Ths latter approah s the moder axomat approah to probablty theory that we shall adopt For a gve expermet, S deotes the sample spae ad 1, 2, 3, represet possble evets We assume that a umber P (, alled the probablty of, satsfes the followg three axoms: (xom 1) P ( 0 for every ; (xom 2) P ( S) 1;

3 Mathematal Statsts 2 3 (xom 3) P ) f j for j 1 1 Example 23-1: Cosder a expermet to toss a far o If we assume that a head s equally to appear as a tal, the we would have P ({ H}) { T}) 1/ 2 Example 23-2: If a de s rolled ad we suppose that all sx sdes are equally lely to appear, the we would have { 1}) {2}) {3}) {4}) {5}) {6}) 1/ 6 From xom 3 t would thus follow that the probablty of rollg a eve umber would equal P ({ 2, 4, 6}) {2}) + {4}) + {6}) 1/ 2 I fat, usg these axoms we shall be able to prove that f a expermet s repeated over ad over aga the, wth probablty 1, the proporto of tmes durg whh ay spefed evet ours wll equal P ( Ths result s ow as the strog law of large umbers Some Propertes of Probablty: (1) For eah S, 1 ) Proof: Se S ad, 1 P ( S) ) + ) (2) The probablty of the ull set s zero; that s P ( ) 0 Proof: Se S, P ( ) 1 S) 0 (3) If ad B are subsets of S suh that B, tha Proof: Se B, B ( ad ( We have + Beause 0, (4) For eah S, 0 P ( 1 Proof: Se S, 0 P ( ) S) 1 (5) (The ddtve Law of Probablty) If ad B are subsets of S, the + Proof: Note that B ( ad B ( ( Se ( ad ( (, + ( ad + Replae by

4 Mathematal Statsts 2 4 (6) If, B, ad C are subsets of S, the B + + B + B Proof: Omtted (7) If 1, 2, 3, are evets, the P ) (Boole s Iequalty) 1 1 Proof: Let B 1 1, B2 2 1, ad geeral B 1 j 1 j It follows that B ad B 1, B2, B3, are mutually exlusve Se B, ( B ) ) P ( 1 2 ) 1 ) + 2 ) + + P Hee, P P B B ) ) smlar result holds for fte uos, e, ) (8) If 1, 2,, are evets, the P 1 ) (Boferro s Iequalty) 1 1 Proof: ppled, together wth P ( 1 ) 1 ) + + ) 1 1 Thus far we have terpreted the probablty of a evet of a gve expermet as beg a measure of how frequey the evet wll our whe the expermet s otually repeated However, there are also other uses of the term probablty Probablty a be terpreted as a measure of the dvdual s belef Furthermore, t seems log to suppose that a measure of belef should satsfy all of the axoms of probablty 24 Codtoal Probablty major objetve of probablty modelg s to determe how lely t s that a evet wll our whe a erta expermet s performed However, there are umerous ases whh the probablty assged to wll be affeted by owledge of the ourree or oourree of aother evet B I suh a example, we wll use the termology odtoal probablty of gve B The otato P ( wll be used to dstgush betwee ths ew oept ad ordary probablty P ( We osder oly those outomes of the radom expermet that are elemets of B; essee, we tae B to be a sample spae, alled the redued sample spae Se B s ow the sample spae, the oly elemets of that oer us are those, f ay, that are also elemets of B, that s, the elemets of B It seems desrable, the, to defe the symbol P ( suh a way that P ( B 1 ad B

5 Mathematal Statsts 2 5 Moreover, from a relatve frequey pot of vew, t would seem logally osstet f we dd ot requre that the rato of the probabltes of the evets B ad B, relatve to the spae B, be the same as the rato of the probabltes of these evets of these evets relatve to the spae S; that s, we should have B B Hee, we have the followg sutable defto of odtoal probablty of gve B P ( Defto 24-1: The odtoal probablty of a evet, gve the evet B, s defed by provded P ( 0 Moreover, we have (1) P ( 0 (2) P ( , provded 1, 2, are mutually exlusve sets (3) P ( B 1 Note that relatve the redued sample spae B, odtoal probabltes defed by above satsfy the orgal defto of probablty, ad thus odtoal probabltes ejoy all the usual propertes of probablty o the redued sample spae Example 24-1: had of 5 ards s to be dealt at radom ad wthout replaemet from a ordary de of 52 playg ards The odtoal probablty of a all-spade had (, relatve to the hypothess that there are at least 4 spades the had (, s, se B, P ( Theorem 24-1 (The Multplatve Law of Probablty): For ay evets ad B, P ( B Example 24-2: bowl otas eght hps Three of the hps are red ad the remag fve are blue Two hps are to be draw suessvely, at radom ad wthout replaemet We wat to ompute the probablty that the frst draw results a red hp ( ad the seod draw results a blue hp ( It s reasoable to assg the followg probabltes:

6 Mathematal Statsts 2 6 P ( 3 / 8 ad P ( 5 / 7 Thus, uder these assgmets, we have P ( (3 / 8)(5 / 7) 15 / 56 Defto 24-2: For some postve teger, let the sets (1) (Mutually Exhaustve) S 2 (2) (Mutually Exlusve) for j 1 j 1, 2, be suh that, The the olleto of sets,,, } s sad to be a partto of S { 1 2 Note that f B s ay subset of S ad { 1, 2,, } s a partto of S, B a be deomposed as follows: B B ) ( B ) ( B ) ( 1 2 Theorem 24-2 (Total Probablty): Suppose that 1, 2,, s a partto of S suh that P ( ) > 0, for 1, 2,, The for ay evet B, Proof: Se the evets ) B ) 1 1 B, 2 B,, B are mutually exlusve, t follows that B 1 ) ad the theorem results from applyg the Multplatve Law of Probablty to eah term ths summato Theorem 24-3 (Bayes Rule): Suppose that 1, 2,, s a partto of S suh that P ( ) > 0, for for 1, 2,, The for ay evet B j ) B j ) j ) B ) 1 Proof: from the defto of the odtoal probablty, ad the Multplatve Theorem, we have j j ) B j ) j The theorem follows by replag the deomator wth the total probablty ) B ) 1 Example 24-3: I a erta fatory, mahe I, II, ad III are all produg sprgs of the same legth Of ther produto, mahe I, II, ad III produe 2, 1, ad 3% defetve sprgs, respetvely Of the total produto of sprgs the fatory, mahe I produes 35%, mahe II produes 25%, ad mahe III produes 40% If oe sprg s seleted at radom from the total

7 Mathematal Statsts 2 7 sprgs produed a day, the probablty that t s defetve, a obvous otato, equals P ( D) I) D I) + II) D II) + III) D III) If the seleted sprg s defetve, the odtoal probablty that t was produed by mahe III s, by Bayes rule, III) D III) P ( III D) D) Note that I, II, ad III are mutually exlusve ad exhaustve evets Example 24-4: I aswerg a questo o a multple-hoe test, a studet ether ows the aswer or guess Let p be the probablty that the studet ows the aswer ad 1 p the probablty that the studet guesses ssume that a studet who guesses at the aswer wll be orret wth probablty 1 m, where m s the umber of multple-hoe alteratves What s the odtoal probablty that a studet ew the aswer to a questo, gve that he or she aswered t orretly? Soluto: Let C ad K deote, respetvely, the evets that the studet aswers the questo orretly ad the evets that he or she atually ows the aswer Now K K K) C K) K) C K) + K ) C K ) p p + (1/ m)(1 mp p) 1 + ( m 1)p Thus, for example, f m 5, p 05, the the probablty that a studet ew the aswer to a questo he or she orretly aswered s 5 / 6 25 Idepedee Defto 25-1: Two evets ad B are alled depedet evets f P ( Otherwse, ad B are alled depedet evets Example 25-1: ard s seleted at radom from a ordary de of 52 playg ards If E s the evet that the seleted ard s a ae ad F s the evet that t s spade, the E ad F are depedet Ths follows beause P ( E F) 1 52, whereas P ( E) 4 52 ad P ( F) Theorem 25-1: If ad B are evets suh that P ( > 0 ad P ( > 0, the ad B are depedet f ad oly f ether of the followg holds: P ( P ( B Note that some textboos use the Theorem 25-1 as the defto of depedet evets

8 Mathematal Statsts 2 8 There s ofte ofuso betwee the oepts of depedet evets ad mutually exlusve evets tually, they are qute dfferet otos, ad perhaps ths s see best by omparsos volvg odtoal probabltes Spefally, f ad B are mutually exlusve, the P ( B 0, whereas for depedet o-ull evets the odtoal probabltes are ozero as oted by Theorem 25-1 I other words, the property of beg mutually exlusve volves a very strog form of depedee, se, for o-ull evets, the ourree of oe evet preludes the ourree of the other evet Theorem 25-2: Two evets ad B are depedet f ad oly f the followg pars of evets are also depedet: B (1) ad (2) ad B (3) ad B Proof: Left as a exerse Defto 25-2: The evets 1, 2,, are sad to be depedet or mutually depedet f for every j 2, 3,, ad every subset of dstt des 1, 2,, r, ) ) ) 1 2 r 1 2 r ) Suppose, B, ad C are three mutually depedet evets, aordg to the defto of mutually depedet evets, t s ot suffet smply to verfy par-wse depedee It would be eessary to verfy P (, P (, P ( B, ad also P ( B Example 25-2: Three evets that are par-wse depedet but ot depedet Cosder the tossg of a o twe, ad defe the three evets,, B, ad C, as follows: s head o the frst toss B s heads o the seod toss C s exatly oe head ad oe tal ( ether order) the two tosses The we have P ( 05 ad B C It follows that ay two of the evets are par-wse depedet However, se the three evets aot all our smultaeously, we have P ( B 0 1 8

9 Mathematal Statsts 2 9 Therefore the three evets,, B, ad C, are ot depedet Example 25-3: Two depedet evets for whh the odtoal probablty of gve B s ot equal to the probablty of Ths surprsg but trval example s possble due to the fat that the odtoal probablty P ( s udefed whe P ( 0 ad so aot be equal to P ( orete example s provded by the hoe S, the etre sample spae, ad B They are depedet beause P ( ) 0 However, beause ths odtoal probablty s ot defed Example 25-4: Idepedet trals, osstg of rollg a par of far de, are performed What s the probablty that a outome of 5 appears before a outome of 7 whe the outome of a roll s the sum of the de? Soluto: If we let E deote the evet that o 5 or 7 appears o the frst 1 trals ad a 5 appears o the th tral, the the desred probablty s P E E ) 1 1 Now, se P{5 o ay tral} 4/36 ad P{7 o ay tral} 6/36, we obta, by the depedee of trals P ( E ) 1 ( ) ( 4 36) ad thus P E ( 13 18) 5 Ths result may also have bee obtaed by usg odtoal probabltes If we let E be the evet that a 5 ours before a 7, the we a obta the desred probablty, E), by odtog o the frst tral, as follows: Let F be the evet that the frst tral results a 5; let G be the evet that t results a 7; ad let H be the evet that the frst tral results ether a 5 or a 7 Se E E F E G E H, we have ( ) ( ) ( ) E) E F ) + E G) + E H ) P ( F) E F) + G) E G) + H ) E H ) However, P ( E F) 1

10 Mathematal Statsts 2 10 P ( E G) 0 P ( E H ) E) The frst two equaltes are obvous The thrd follows beause, f the frst outome results ether a 5 or a 7, the at that pot the stuato s exatly as whe the problem frst started; amely, the expermeter wll otually roll a par of far de utl ether a 5 or 7 appears Furthermore, the trals are depedet; therefore, the outome of the frst tral wll have o effet o subsequet rolls of the de Se 4 P ( F) 36 we see that P ( G) P ( H ), P ( E) + E) Note that the aswer s qute tutve That s, se a 5 our o ay roll wth probablty 4/36 ad a 7 wth probablty 6/36, t seems tutve that the odds that a 5 appears before a 7 should be 6 to 4 agast The probablty should be 4/10, as deed t s The same argumet shows that f E ad F are mutually exlusve evets of a expermet, the, whe depedet trals of ths expermet are performed, the E wll our before the evet F wth probablty E) E) + F)